Doing Sums Anthony G. OFarrell National University of Ireland - PowerPoint PPT Presentation

Doing Sums Anthony G. O’Farrell National University of Ireland Maynooth University of Washington, Seattle, August 2019

1968: On Pennsylvania Avenue

1975: Gamelin’s VII

1974-5: UCLA-Caltech: Gamelin’s XI

2004: Don+Marianne, Brú na Bóinne:

2004: Monasterboice:

Doing sums Beautiful theorems Good open problems

Sums of Algebras If X is a set and F a field, then F X is an algebra over F , when endowed with pointwise operations. We shall consider F = C and F = R .

Sums of Algebras If X is a set and F a field, then F X is an algebra over F , when endowed with pointwise operations. We shall consider F = C and F = R . F [ f ] is the subalgebra generated by 1 and f , i.e. the set of all functions x �→ a 0 + a 1 · f ( x ) + · · · + a n · f ( x ) n where each a j ∈ F .

Sums of Algebras If X is a set and F a field, then F X is an algebra over F , when endowed with pointwise operations. We shall consider F = C and F = R . F [ f ] is the subalgebra generated by 1 and f , i.e. the set of all functions x �→ a 0 + a 1 · f ( x ) + · · · + a n · f ( x ) n where each a j ∈ F . F [ f , g ] is the subalgebra generated by 1, f , and g .

Sums of Algebras If X is a set and F a field, then F X is an algebra over F , when endowed with pointwise operations. We shall consider F = C and F = R . F [ f ] is the subalgebra generated by 1 and f , i.e. the set of all functions x �→ a 0 + a 1 · f ( x ) + · · · + a n · f ( x ) n where each a j ∈ F . F [ f , g ] is the subalgebra generated by 1, f , and g . F [ f ] + F [ g ] is the set of all sums h + k , with h ∈ F [ f ] and k ∈ F [ g ] . And so on.

Theorem (1974) Let ϕ and ψ be homeomorphisms of C into C having opposite degrees. Let Y ⊂ C be compact, C \ Y be connected, and X = bdy ( Y ) . Then C [ ϕ ] + C [ ψ ] is dense in C ( X , C ) .

Theorem (1974) Let ϕ and ψ be homeomorphisms of C into C having opposite degrees. Let Y ⊂ C be compact, C \ Y be connected, and X = bdy ( Y ) . Then C [ ϕ ] + C [ ψ ] is dense in C ( X , C ) .

Theorem (Walsh 1929, ‘Walsh-Lebesgue’) Suppose X ⊂ C is compact, the unbounded component Ω of C \ X is connected, and X = bdy (Ω) . Then { u ∈ R [ x , y ] : ∆ u = 0 on R 2 } is dense in C ( X , R ) . i.e. C [ z ] + C [¯ z ] is dense in C ( X , C ) . Theorem (Browder-Wermer 1964) Let ψ : S 1 → S 1 be a direction-reversing homeomorphism. Then every continuous function on S 1 can be uniformly approximated by linear combinations of powers z n , n ≥ 0 , and ψ n , n ≥ 0 , i.e. C [ z ] + C [ ψ ] is dense in C ( S 1 , C ) .

Theorem (+Preskenis, 1984) Let k ∈ N . Let ϕ and ψ be C k diffeomorphisms of C into C having opposite degrees. Let X ⊂ C be compact. Then C [ ϕ, ψ ] is dense in C k ( X , C ) . Question: What about mere homeomorphisms?

Garnett Theorem (+Garnett, 1976) Let ǫ > 0 and ψ be a C 1 + ǫ direction-reversing involution of S 1 onto S 1 . Then C [ ϕ ] + C [ ψ ] is dense in C 1 ( S 1 , C ) .

Example (+Garnett, 1976) There exists a direction-reversing involution ψ on S 1 of class W 1 , 1 such that C [ z ] + C [ ψ ] is not dense in W 1 , 1 ( S 1 , C ) . This figure has other uses!

Theorem (1986) C ∞ maps can increase C ∞ dimension. There exists a C ∞ function from R → R 2 that has image of C ∞ dimension 2.

Also answers another question: A := clos C ( S 1 ) C [ z ] , the disk algebra on the circle. For a homeomorphism ψ of S 1 onto S 1 , let A ψ := A ∩ A ◦ ψ . Theorem (Browder+Wermer, 1964) If ψ is a singular homeomorphism of the circle onto itself, then Re A ψ is dense in C ( S 1 /ψ, R ) . Question: When is A ψ trivial, i.e. consisting only of constants? Wermer asked me whether it might hold when ψ is absolutely-continuous. This is not so, and it remains unclear.

Marshall Going his way:

Lunchtime conversation: a trip

Theorem (+Marshall, 1979) Let X ⊂ R 2 be compact, and suppose all orbits are closed. Then R [ x ] + R [ y ] is dense in C ( X , R ) if and only if there are no round trips in X.

Theorem (+Marshall, 1979) Let X ⊂ R 2 be compact, and suppose all orbits are closed. Then R [ x ] + R [ y ] is dense in C ( X , R ) if and only if there are no round trips in X. Same for A 1 + A 2 , for any two subalgebras of any C ( X ) that contain the constants.

Theorem (+Marshall, 1979) Let X ⊂ R 2 be compact, and suppose all orbits are closed. Then R [ x ] + R [ y ] is dense in C ( X , R ) if and only if there are no round trips in X. Same for A 1 + A 2 , for any two subalgebras of any C ( X ) that contain the constants. Big subject: Trips become lightning bolts. Havinson’s example.

µ n , ( b 1 , b 2 ,... ) := 1 n ( b 1 − b 2 + b 3 − b 4 + · · · ± b n ) . Theorem (+Marshall, 1983) Let X ⊂ R 2 be compact. Then R [ x ] + R [ y ] is dense in C ( X , R ) if and only if µ n , b → 0 weak-star for each lightning-bolt b in X. Proof. Focus on an extreme norm 1 annihilator, and apply Birkhoff’s Ergodic Theorem to the right map on the space of bolts.

More than 2 algebras The difference between sums of two algebras and sums of three, or more. Hilbert’s 13th problem. Kolmogoroff. Arnold. Vitushkin.

Yaki Sternfeld Theorem (Sternfeld, 1985) Let 2 ≤ n ∈ N , and let X be a compact metric space. Then the following are equivalent: (1) X has topological dimension n. (2) ∃ ϕ i ∈ C ( X , R )( i = 1 , . . . 2 n + 1 ) such that each f ∈ C ( X ) is representable as 2 n + 1 � f ( x ) = g i ( ϕ i ( x )) i = 1 with each g i ∈ C ( R ) . (3) X is homeomorphic to a compact set Y ∈ R 2 n + 1 such that on Y we have: R [ x 1 ] + · · · + R [ x 2 n + 1 ] = C ( Y , R ) .

Example (Sternfeld, 1986): There exists a compact set X ⊂ R 3 such that each bounded function f ∈ R X is expressible as as sum f = g 1 ( x ) + g 2 ( y ) + g 3 ( z ) , with bounded functions g i ∈ R R , but there exists a continuous function f that cannot be so represented using continuous (or even Borel) functions g i .

The culprit: F 2 .

Reversibility

THANKS